ATD: Anomaly detection and functional data analysis with applications to threat detection for multimodal satellite data

海量数据分析和人工智能（AI）的进步相结合，正在给社会带来重大变化。这一点至关重要的一个领域是遥感和地理信息系统（GIS）（如卫星数据）的整合，这对于了解人类活动对环境和气候的影响非常重要。森林砍伐是气候变化的一个关键因素，对生态系统、生物多样性和人口造成负面影响。幸运的是，现在有大量的卫星数据集可以帮助检测森林砍伐，特别是在亚马逊这样的关键森林。然而，由于云层和阴影等因素，跟踪森林砍伐、退化和森林再生具有挑战性。为了应对这些挑战，研究人员将开发一种基于新的数学框架的创新方法。本研究涉及人类迁徙、气候变化、交通物流、传染病传播等多个领域。调查结果将对情报收集有价值，有助于了解安全状况，为评估和决策提供信息，包括那些具有军事和政治影响的评估和决策。此外，研究人员致力于培训和培养学生在这些领域的专业知识，为他们提供宝贵的学习机会。研究人员将在数学泛函分析的框架内，利用Karhunen-Loeve （KL）展开等表示，开发一种创新和独特的方法来进行变点和异常检测。这种方法在几个重要方面偏离了以前的方法。KL展开是表示随机过程的理想选择，它提供了最优表示。它们表现出显著的通用性，能够在复杂的几何域上准确地表示各种过程和领域。检测是通过构造和匹配为截断的KL展开量身定制的嵌套特征空间来实现的。与现有的统计方法不同，该方法基于功能分析，在检测复杂领域的隐藏现象方面具有以下优势：1)有原则地检测异常的全局和局部信号。2)使用稳健的浓度不等式开发可靠的假设检验，而不假设数据分布（必不可少），例如检测的稳健统计显著性。3)精确的异常量化。4)适用于不同的几何形状，包括地理空间、时空、曲面、网络等。嵌套子空间的构造涉及到从计算应用数学和高性能计算中衍生出来的高效算法。5)集成多级别过滤器，能够以近乎最佳的性能处理大量数据。总的来说，这种基于功能分析的方法为变化和异常检测提供了一个新的视角。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The combination of massive data analysis and advancements in Artificial Intelligence (AI) is causing major changes in society. One area where this is crucial is the integration of remote sensing and Geographic Information System (GIS), like satellite data, which is important for understanding the impact of human activities on the environment and climate. Deforestation is a key factor in climate change, causing negative effects on ecosystems, biodiversity, and human populations. Fortunately, there are now extensive satellite datasets that can help detect deforestation, especially in critical forests like the Amazon. However, tracking deforestation, degradation, and forest regrowth is challenging due to factors like clouds and shadows. To address these challenges, the investigators will develop an innovative approach based on a new mathematical framework. This research is relevant to various fields such as human migration, climate change, transportation logistics, and epidemic disease diffusion. The findings will be valuable for intelligence gathering and will contribute to understanding security conditions, informing assessments and decisions, including those with military and political implications. Moreover, the investigators are committed to training and nurturing students' expertise in these areas, providing them with valuable learning opportunities.The investigators will develop an innovative and distinct approach to change-point and anomaly detection within the framework of mathematical functional analysis, utilizing representations like the Karhunen-Loeve (KL) expansion. This approach deviates from previous methods in several significant ways. KL expansions are ideal for representing random processes, providing optimal representations. They exhibit a remarkable level of generality, enabling accurate representation of various processes and fields over complex geometrical domains. Detection is achieved by constructing and matching nested eigenspaces tailored to truncated KL expansions. Unlike current statistical approaches, the proposed approach is rooted in functional analysis and offers several advantages for detecting hidden phenomena in complex domains: 1) Principled detection of anomalous global and local signals. 2) Development of reliable hypothesis tests using robust concentration inequalities without making assumptions about data distributions (essential) e.g. robust statistical significance for detection. 3) Precise anomaly quantification. 4) Applicability to diverse geometries, including geo-spatial, spatio-temporal, surfaces, networks, etc. The construction of nested subspaces involves efficient algorithms derived from computational applied mathematics and high-performance computing. 5) Integration of multilevel filters capable of processing massive volumes of data with near-optimal performance. Overall, this approach, rooted in functional analysis, presents a new perspective on change and anomaly detection.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.